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• Operand

Operand-equiv

Basic equivalence relation for operand structures.

Definitions and Theorems

Function: operand-equiv$inline

(defun operand-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (operandp acl2::x)
                              (operandp acl2::y))))
  (equal (operand-fix acl2::x)
         (operand-fix acl2::y)))

Theorem: operand-equiv-is-an-equivalence

(defthm operand-equiv-is-an-equivalence
  (and (booleanp (operand-equiv x y))
       (operand-equiv x x)
       (implies (operand-equiv x y)
                (operand-equiv y x))
       (implies (and (operand-equiv x y)
                     (operand-equiv y z))
                (operand-equiv x z)))
  :rule-classes (:equivalence))

Theorem: operand-equiv-implies-equal-operand-fix-1

(defthm operand-equiv-implies-equal-operand-fix-1
  (implies (operand-equiv acl2::x x-equiv)
           (equal (operand-fix acl2::x)
                  (operand-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: operand-fix-under-operand-equiv

(defthm operand-fix-under-operand-equiv
  (operand-equiv (operand-fix acl2::x)
                 acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-operand-fix-1-forward-to-operand-equiv

(defthm equal-of-operand-fix-1-forward-to-operand-equiv
  (implies (equal (operand-fix acl2::x) acl2::y)
           (operand-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-operand-fix-2-forward-to-operand-equiv

(defthm equal-of-operand-fix-2-forward-to-operand-equiv
  (implies (equal acl2::x (operand-fix acl2::y))
           (operand-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: operand-equiv-of-operand-fix-1-forward

(defthm operand-equiv-of-operand-fix-1-forward
  (implies (operand-equiv (operand-fix acl2::x)
                          acl2::y)
           (operand-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: operand-equiv-of-operand-fix-2-forward

(defthm operand-equiv-of-operand-fix-2-forward
  (implies (operand-equiv acl2::x (operand-fix acl2::y))
           (operand-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)